No visa stamping is done in the US. You must visit a consulate/embassy outside the US. A visa is only for entry. If your L1A status has been applied as a change of status and you receive an I-94 with the approval, you will be in L1A status from the date on the I-94. If COS was not approved, you must go for stamping and reenter to be able to work on L1A.

You are a long way from a green card. Your statement that you do not want to leave the US before getting a green card is strange. Why are you worried about traveling in another status? Remaining in the US does nothing to speed up the process or guarantee success.